The Mathematics of Arbitrage

15.1 Introduction 323

tends to zero and such that the sequence of stopped processes

(

(Mn)Tn

)

n≥ 1 is
relatively weakly compact inH^1.
If all the martingales are of the formMn=Hn·Mfor a fixed continuous
local martingale taking values inRd, then the elements in theH^1 -closed convex
hull of the sequence

(

(Mn)Tn

)

n≥ 1 are also of the formH·M.

As a consequence we obtain the existence of convex combinations

Kn∈conv{Hn,Hn+1,...}

such thatKn (^1) [[ 0,Tn]]·Mtends to a limitH^0 ·MinH^1. Also remark that the
remainingsingularpartsKn (^1) ]]Tn,∞]]·Mtend to zero in a stationary way, i.e.
for almost eachω∈Ωtheset{t|∃n≥n 0 ,Ktn=0}becomes empty for large
enoughn 0. As a result we immediately derive that the sequenceKn·Mtends
toH^0 ·Min the semi-martingale topology.
If the local martingaleMis not continuous the situation is more delicate.
In this case we cannot obtain a limit of the formH^0 ·Mand also the decom-
position is not just done by stopping the processes at well-selected stopping
times.
Theorem 15.B.LetM be anRd-valued local martingale and(Hn)n≥ 1 be
asequenceofM-integrable predictable processes such that(Hn·M)n≥ 1 is an
H^1 bounded sequence of martingales.
Then there is a subsequence, for simplicity still denoted by(Hn)n≥ 1 ,an
increasing sequence of stopping times(Tn)n≥ 1 , a sequence of convex combi-
nationsLn=

∑

k≥nα

n

kH

kas well as a sequence of predictable sets(En)n≥ 1

such that

(1)En⊂[[ 0,Tn]]andTnincreases to∞,
(2)the sequence

(

Hn (^1) [[ 0,Tn]]∩(En)c·M

)

n≥ 1 is weakly relatively compact inH

(^1) ,
(3)

∑

n≥ 11 En≤d,
(4)the convex combinations

∑

k≥nα

k

nH

k (^1) [[ 0,T
n]]∩(En)c·Mconverge inH
(^1) to
a stochastic integral of the formH^0 ·M, for some predictable processH^0 ,
(5)the convex combinationsVn=

∑

k≥nα

k
nH
k 1
]]Tn,∞[[∪En·Mconverge to
acadl ag optional processZof finite variation in the following sense: a.s.
we have thatZt= lims↘t;s∈Qlimn→∞(Vn)sfor eacht∈R+,
(6)the brackets[(H^0 −Ln)·M,(H^0 −Ln)·M]∞tend to zero in probability.

If, in addition, the set

{∆(Hn·M)−T|n∈N;Tstopping time}

resp.
{|∆(Hn·M)T||n∈N;Tstopping time}

is uniformly integrable, e.g. there is an integrable functionw≥ 0 such that

∆(Hn·M)≥−w resp. |∆(Hn·M)|≤w, a.s.

then the process(Zt)t∈R+is decreasing (resp. vanishes identically).